Atmel Software Framework

math Namespace Reference

Data Structures

class  matrix3d
 3-Dimensional Matrix Class More...
 
class  matrix4d
 4-Dimensional Matrix Class More...
 
class  plane
 3-dimensional plane class declaration More...
 
class  quaternion
 4-dimensional normed division algebra over the Real Numbers More...
 
class  vector2d
 Ordered 2-tuple implementing Euclidean vector operations. More...
 
class  vector3d
 Ordered 3-tuple implementing Euclidean vector operations. More...
 
class  vector4d
 Ordered 4-tuple implementing Euclidean vector operations. More...
 
class  vector4h
 Augmented 3-dimensional vector / homogeneous vector / projected point. More...
 

Functions

scalar det (scalar a, scalar b, scalar c, scalar d)
 Calculate the determinant of a 2x2 matrix. More...
 
scalar det (scalar a1, scalar b1, scalar c1, scalar a2, scalar b2, scalar c2, scalar a3, scalar b3, scalar c3)
 Calculate the determinant of a 3x3 matrix. More...
 
const quaternion operator* (const quaternion &q, const vector3d &v)
 Multiply a vector by a quaternion. More...
 
const quaternion operator* (const vector3d &v, const quaternion &q)
 Multiply a quaternion by a vector. More...
 
const matrix3d quaternion2DCM (const quaternion &q)
 Quaternion to direction cosine matrix (DCM) conversion. More...
 
const vector3d quaternion2euler (const quaternion &q)
 Quaternion to Euler angle conversion. More...
 
const vector3d rotate (const quaternion &q, const vector3d &v)
 Rotate a vector by a unit quaternion. More...
 

scalar math::det ( scalar  a,
scalar  b,
scalar  c,
scalar  d 
)
inline

Calculate the determinant of a 2x2 matrix.

This routine calculates the determinant of a 2x2 matrix consisting of entries a, b, c, and d arranged in the matrix as follows:

\[ \left| \begin{array}{cc} a & b \\ c & d \end{array} \right| \]

Return values
scalarThe real-valued determinant of the input matrix.

Referenced by math::matrix3d::adjoint(), math::matrix4d::adjoint(), math::vector3d::cross(), math::vector4h::cross(), det(), and math::matrix4d::determinant().

scalar math::det ( scalar  a1,
scalar  b1,
scalar  c1,
scalar  a2,
scalar  b2,
scalar  c2,
scalar  a3,
scalar  b3,
scalar  c3 
)
inline

Calculate the determinant of a 3x3 matrix.

This routine calculates the determinant of a 3x3 matrix consisting of entries arranged in the matrix as follows:

\[ \left| \begin{array}{ccc} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{array} \right| \]

Return values
scalarThe real-valued determinant of the input matrix.

References det().

const quaternion math::operator* ( const quaternion &  q,
const vector3d &  v 
)
inline

Multiply a vector by a quaternion.

This operator multiplies a math::vector3d object on the right-side of the operator by math::quaternion object on the left-side of the operator.

Parameters
qA math::quaternion operand.
vA math::vector3d operand.
Return values
math::quaternionThe product of the input operands.
const quaternion math::operator* ( const vector3d &  v,
const quaternion &  q 
)
inline

Multiply a quaternion by a vector.

This operator multiplies a math::quaternion object on the right-side of the operator by math::vector3d object on the left-side of the operator.

Parameters
vA math::vector3d operand.
qA math::quaternion operand.
Return values
math::quaternionThe product of the input operands.
const matrix3d math::quaternion2DCM ( const quaternion &  q)
inline

Quaternion to direction cosine matrix (DCM) conversion.

This routine converts a specified quaternion \( \mathbf{q} \) to a 3x3 direction cosine matrix (DCM).

If point \( \mathbf{P} \) is transformed to \( \mathbf{P^{\prime}} \) as a result of a rotation described by quaternion \( \mathbf{q} \) according to the following quaternion algebra:

\( P^{\prime} = qPq^{*} \)

where

\( q = q_0 + \hat{i}q_1 + \hat{j}q_2 + \hat{k}q_3 \)

\( q^{*} = q_0 - \hat{i}q_1 - \hat{j}q_2 - \hat{k}q_3 \)

\( P = 0 + \hat{i}x + \hat{j}y + \hat{k}z \)

Parameters
qA rotation math::quaternion argument.
Return values
math::matrix3dA 3x3 direction cosine matrix.
const vector3d math::quaternion2euler ( const quaternion &  q)
inline

Quaternion to Euler angle conversion.

This routine converts a specified quaternion \( \mathbf{q} \) to a 3x1 vector storing bank (rotation about x-axis), attitude (rotation about y-axis), and heading (rotation about z-axis) values in the x, y, and z vector elements, respectively.

Note
  • Singularities are not accounted for when the pitch approaches \( \pm 90 \) degrees (north / south pole); i.e. gimbal lock.
  • This is a special-case assuming a particular right-hand system, but there are many types of Euler angles depending upon the system being described. Additional work is required to create a set of useful utility methods for converting quaternions to Euler angles.
Parameters
qA rotation math::quaternion argument.
Return values
math::vector3dA 3x1 vector storing roll, pitch, and yaw.
const vector3d math::rotate ( const quaternion &  q,
const vector3d &  v 
)
inline

Rotate a vector by a unit quaternion.

Given unit quaternion \( q = (s, \vec{v}) \) and its conjugate \( q^{*} = (s, -\vec{v}), \) to rotate arbitrary vector \( \vec{r} \) by an angle \( \alpha \) about a vector \( \hat{N} \):

  • Convert the vector \( \vec{r} \) to quaternion form \( p = (0, \vec{r}) \) and express the rotation quaternion as:

\[ q = (\cos{(\frac{\alpha}{2})}, \sin{(\frac{\alpha}{2})}\hat{N}) \]

  • Perform the following multiplication which is equivalent to rotation about \( \hat{N} \):

\[ qpq^{*} = (0, (s^{2} - \vec{v} \cdot \vec{v})\vec{r} + 2\vec{v}(\vec{v} \cdot \vec{r}) + 2s\vec{v} \times \vec{r}) = (0, \vec{r}^{\prime}) \]

Parameters
qA math::quaternion operand.
vA math::vector3d operand.
Return values
math::vector3dThe resulting rotated vector object.